Being a Cauchy sequence is not a topological property, i.e. two metrics can induce the same topology and yet a sequence which is Cauchy in one may not be Cauchy in the other. It is a uniform property though, i.e. if two metrics induce the same uniformity then they have the same set of Cauchy sequences. But I'm wondering if Cauchy sequences can be defined in weaker conditions than a uniform space.
Let $X$ be a topological space endowed with a bornology, i.e. a structure which defines a notion of bounded sets. My question is, is it possible to define the notion of Cauchy sequences in terms of this bornology? To put it another way, if two metrics induce both the same topology and the same bornology, then do they have the same set of Cauchy sequences?
Best Answer
Consider the function $f:(0,1)\to\mathbb{R}$ defined by $f(x)=\sin(\frac1x)$. Then $x\mapsto(x,f(x))$ is a homeomorphism from $(0,1)$ to the graph of $f$. However, $a_n=\frac2{n\pi}$ is a Cauchy sequence in $(0,1)$, while $f(a_n)$ is not a Cauchy sequence in the graph of $f$, since the $y$ values form the divergent sequence $0,1,0,-1,0,1,0,-1...$. Moreover, both $(0,1)$ and the graph of $f$ are bounded metric spaces, so they are not only homeomorphic, but also 'bornoleomorphic'.